 So, we have this definition for the Gibbs free energy. You may be more comfortable thinking of as the enthalpy minus T times the entropy or you can think of it if you prefer as the Helmholtz energy plus a PV product. All these say the same thing. They give us a definition for the Gibbs energy. To make that useful, what we'll want to do often is pay attention to how the Gibbs free energy changes as one of these other thermodynamic variables is changing. So, let's write down the fundamental equation for the Gibbs energy. Let's take the differential of this expression. So, differential of G is DG. On the right side, differential of U is DU. PV becomes PDV plus VDP. If I take the differential of it and this negative TS terms becomes minus TDS and a minus SDT and since we know a fundamental equation for DU. DU can itself be written as TDS minus PDV. So, if I use that term for DU and then rewrite the rest of these terms. I've got PDV plus VDP minus TDS minus SDT. There's now quite a bit of cancellation that happens. This TDS term cancels a minus TDS term. I have a minus PDV term that's going to cancel a positive PDV term, leaving us with just two of these when we're done. And if I switch the order of those two, I'll write this fundamental equation for DG as minus s times the change in temperature. Plus volume times the change in pressure. So, that form should be looking fairly familiar by now. Differential change in some flavor of the energy is some thermodynamic variables multiplying some changes in other thermodynamic variables. And what this fundamental equation tells us immediately is two things. It tells us the natural variables of the function. G, when thought of as a function of T and P, has a relatively simple form. So, as we hope to find when we define the Gibbs free energy, we've now seen that the natural variables of the Gibbs free energy are indeed temperature and pressure. The other thing we can see immediately from looking at the fundamental equation is some of the thermodynamic derivatives. If I would like to know how quickly the Gibbs free energy is changing as I change the temperature, DG, DT, as long as I'm holding pressure constant so that this term disappears, DG, DT at constant P is this coefficient, negative s. And likewise, DG, DT at constant temperature is this coefficient volume. So, I'll go ahead and put those equations in a box because we will use those two in particular quite often because the Gibbs free energy turns out to be very important in telling us about the spontaneity of a process, as we'll see when the temperature and pressure are held constant. So, if we want to know what processes spontaneously take place, these expressions are going to tell us quite a bit. So, we'll use these expressions often. So, make sure and remember those. And we can move on next and calculate things like the thermodynamic connection formula for the Gibbs free energy.